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Octahedron problem

  1. Jan 29, 2009 #1
    If each side of a tetrahedron is an equilateral triangle painted white or black, five distinct color patterns are possible: all sides white, all black, just one side white and the rest black, just one side black and the rest white, and two sides white while the other two are black. If each side of an octahedron is an equilateral triangle painted white or black, how many distinct patterns are possible?

    I was able to come up with a good answer but I'm not sure if its right or not.
     
  2. jcsd
  3. Jan 29, 2009 #2
    3!.3!.28/26 I think...

    The symmetry is the catch here...
     
  4. Jan 29, 2009 #3

    tiny-tim

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    Welcome to PF!

    Hi reese92tsi! Welcome to PF! :smile:
    Show us (with your reasoning, of course)! :wink:
     
  5. Jan 29, 2009 #4
    Re: Welcome to PF!

    Thanks for the warm welcome

    all sides black: 1 pattern
    all white: 1

    7 white 1 black: 1
    7 black 1 white: 1

    6 white 2 black: 3
    6 black 2 white: 3
    -2 white faces adjacent at an edge
    -2 white faces adjacent at a vertex
    -2 white faces neither adjacent at an edge or vertex

    5 white 3 black: 3
    5 black 3 white: 3
    -3 white faces adjacent at edges
    -3 white faces adjacent at only vertices
    -2 white faces adjacent at an edge and 1 white face
    adjacent at either vertex of the set of the 2 edge adjacent white faces

    4 black 4 white: 7
    -4 white faces sharing a common vertex
    -3 white faces sharing a common vertex and 1 white face
    adjacent at only the vertex of the rightmost and leftmost
    white face of the 3
    -3 white faces sharing a common vertex and 1 white face
    adjacent to only the edge of the right most face of the 3
    -The mirror to the pattern above
    -3 white faces sharing a common vertex and 1 white face adjacent to
    only the edge of the center white face of the 3
    -4 white faces adjacent only at vertices
    -2 sets of 2 white faces which are adjacent at an edge
    touching each other at only 2 vertices

    So I've got 23 patterns
     
  6. Jan 29, 2009 #5

    tiny-tim

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    Hi reese92tsi! :smile:

    0 1 2 and 4 white look good, but I'm not convinced about 3 …

    what does 3 white faces adjacent at edges mean?

    and have you checked that there are no mirror images left out?
     
  7. Jan 29, 2009 #6
    I'm sorry about the phrasing. "3 white faces sharing a common vertex" is probably a better way of phrasing it.

    Going over the patterns in my head it looks like I've found them all (still not positive).

    Can you explain this formula in further detail, I'm not really a math buff (would love to become one). I think puzzles/logic problems are a lot of fun.
     
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